(2x+3y)^4

2 min read Jun 16, 2024

Expanding (2x + 3y)^4 using the Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions of the form (x + y)^n. It states that:

(x + y)^n = Σ (n choose k) x^(n-k) y^k

where (n choose k) represents the binomial coefficient, which is calculated as:

(n choose k) = n! / (k! * (n-k)!)

Let's apply this to our expression (2x + 3y)^4:

We have x = 2x and y = 3y, and n = 4.

We need to calculate the binomial coefficients for k = 0, 1, 2, 3, and 4:

(4 choose 0) = 4! / (0! * 4!) = 1
(4 choose 1) = 4! / (1! * 3!) = 4
(4 choose 2) = 4! / (2! * 2!) = 6
(4 choose 3) = 4! / (3! * 1!) = 4
(4 choose 4) = 4! / (4! * 0!) = 1

Now we can substitute these coefficients and the terms into the binomial theorem formula:

(2x + 3y)^4 = 1 * (2x)^4 * (3y)^0 + 4 * (2x)^3 * (3y)^1 + 6 * (2x)^2 * (3y)^2 + 4 * (2x)^1 * (3y)^3 + 1 * (2x)^0 * (3y)^4

Finally, let's simplify the expression by evaluating the powers and multiplying:

**(2x + 3y)^4 = ** 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4

Therefore, the expanded form of (2x + 3y)^4 is 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4.